So now we know the definition of
Multivariate function and know how to draw
some understanding of it by
administering the levels of
its function to the graph, right?
So now let's talk about
the limits of the multivariate function.
Let us just stick with
our basic understanding of what the limit is.
The limit is the value that
our function resembles the most.
Basically, what it means for our multivariate case,
in our multivariate case we can just
generalize our extremely complicated
but a useful definition
with Epsilon Delta technique, right?
So what have we been saying in it?
We are basically saying that the way you
see is called Functions Limit.
In some for example, two-dimensional points whatsoever.
If for any given deviation we can go as close to
limit point that function do not deviate
from our limits sufficiently,
but much more than our undefined deviation, right?
Okay. That's basically what it is and it doesn't
change at all but as a phrase,
we can go as close to our limit point
implies that we now can
say what is close to how limit point and what is not.
So basically what we're talking about;
we're talking about our point a b,
we are talking about some arbitrary point x y,
and we're going to say is that the distance between
these two points are smaller than our,
for example, our delta z.
It is delta neighborhood where in
the close neighborhood of our limit point.
In order to do it, we need to understand some basics.
For example, how to calculate
the distance between point x and the y,
and a and b.
In order to do it, we're just going to use
straight-out pythagoras theorem which is well,
you don't know what to do, use pythagoras theorem.
So our distance here is basically square root from
sum of squares of the sides of this triangle.
Which means that the square root from x minus a
squared plus y minus b squared.
Okay. That's nice but how does
it differ from the case of single-headed functions?
From the standpoint of definition,
it's pretty much the same.
But let us consider for example our real plan
and we are looking at some of our limits point ab.
So what should we expect?
Previously what we've talked about is that
we're just trying to get as close
to our limit point from
all possible straight directions, right?
You do remember we had two definitions,
one complicated with epsilon delta
and one with sequences, right?
So sequences basically told us go
to the limit points as close as you can and
then the limit of the sequence of actions failures
should approach all the same value c, right?
Because this can actually be
easily extrapolated here but,
and this is a very big but here.
Now they're considering all approaches on the plane,
not on our green straight line or our real numbers.
Okay? So now we can go for example,
from the left, from the right as in single variate case.
But also vertically from
the top or by some straight line here or maybe
just by parabola or just by
some random points into the plan.
So the only rule here we need to
approach our limit point ab.
Okay? So this is extremely more
complicated because we have
additional degree of freedom, additional variable y.
Thus we need to consider
much more possible approaches that we
can use to get to the limit function, limit point.
More importantly,
the idea is that we need just to consider limit
by one variable hands-on
but the other will not suffice by
the very idea that there's
only one way to get here, right?
Lets write english way.
Firstly, you need to throw
yourself on to the for example,
x axis or y axis
and then you need to consider only one variable.
This is only two approaches,
we need to consider them all.
So let us revise what we do have here.
Since we've established that we need to consider
all approaches to prove
for example or calculate our limits,
we need to prove that on all approaches,
we get the same results.
It's nightmarish.
So the technique we're going to
use primarily in order to calculate
function limits is basically to
truncate this calculation to
the case of single variables.
For example, change all the calculations we're doing into
the case of single variate functions
by substituting variables,
by grouping variables, by just subtracting and
just turning our function
into the pieces of single-variate functions at all.
Okay, So but to prove the absence of the limit,
it's quite nice to consider that we can
just demonstrate a pair of approaches.
For example, horizontal or vertical one or some others
with two different limits and
so the limit itself won't exist.
So for example, let
us take a look at basic polynomial functions.
Do not be disturbed by the last term.
As you can see, it's basically a polynomial
just we can get rid
out of brackets and thus we
will get pretty nice polynomial function.
How can one say that
the limit of this function is easy to calculate?
Because it should be easy to calculate, let's face it.
What if the polynomial function
is not easy to calculate, right?
First of all, all arithmetic rules
are the same because the definition is the same, right?
Thus, we should consider only the limits
of the terms separable right?
So in order to consider
the limits of this term for example,
let us take the term five y powered five multiplied by x.
Okay? Why it is
easily truncatable to the case of single varied function?
Because it's easier as
a product of single varied at function
y equals to function equals to five,
function equals 2y powered five.
Its single-varied function towards y, right?
As a product of function
x which is single varied function towards x.
Thus it approaches, five approaches five,
y approaches one thus
y powered five approaches
one power five and 10x approaches too.
This term approaches ten.
Okay? That's easy and thus we can calculate
the limit of this polynomial just variable vice.
To proceed further, we will cancel with
some more complicated examples in the following video.